A294675 -id:A294675 - cp's OEIS frontend

A294741 Numbers that are the sum of 5 nonzero squares in exactly 7 ways.

77, 83, 85, 88, 94, 99, 120, 124, 130, 137, 138, 150, 156, 201

Offset: 1

Robert Price, Nov 07 2017

Theorem: There are no further terms. Proof (from a proof by David A. Corneth on Nov 08 2017 in A294736): The von Eitzen link states that if n > 5408 then the number of ways to write n as a sum of 5 squares is at least floor(sqrt(n - 101) / 8) = 9. For n <= 5408, terms have been verified by inspection. Hence this sequence is finite and complete.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A025429, A025357, A294675, A294736.

fQ[n_] := Block[{pr = PowersRepresentations[n, 5, 2]}, Length@Select[pr, #[[1]] > 0 &] == 7]; Select[Range@250, fQ] (* Robert G. Wilson v, Nov 17 2017 *)

A294742 Numbers that are the sum of 5 nonzero squares in exactly 8 ways.

Original entry on oeis.org

91, 104, 106, 119, 122, 123, 126, 141, 143, 162, 185, 225

Offset: 1

Robert Price, Nov 07 2017

Theorem: There are no further terms. Proof (from a proof by David A. Corneth on Nov 08 2017 in A294736): The von Eitzen link states that if n > 5408 then the number of ways to write n as a sum of 5 squares is at least floor(sqrt(n - 101) / 8) = 9. For n <= 5408, terms have been verified by inspection. Hence this sequence is finite and complete.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A025429, A025357, A294675, A294736.

fQ[n_] := Block[{pr = PowersRepresentations[n, 5, 2]}, Length@Select[pr, #[[1]] > 0 &] == 8]; Select[Range@250, fQ] (* Robert G. Wilson v, Nov 17 2017 *)

A294743 Numbers that are the sum of 5 nonzero squares in exactly 9 ways.

Original entry on oeis.org

101, 112, 115, 118, 127, 144, 159, 161, 165, 169, 180

Offset: 1

Robert Price, Nov 07 2017

Theorem: There are no further terms. Proof (from a proof by David A. Corneth on Nov 08 2017 in A294736): The von Eitzen link states that if n > 6501 then the number of ways to write n as a sum of 5 squares is at least 10. For n <= 6501 terms have been verified by inspection. Hence this sequence is finite and complete.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A025429, A025357, A294675, A294736.

fQ[n_] := Block[{pr = PowersRepresentations[n, 5, 2]}, Length@Select[pr, #[[1]] > 0 &] == 9]; Select[Range@250, fQ](* Robert G. Wilson v, Nov 17 2017 *)

A294744 Numbers that are the sum of 5 nonzero squares in exactly 10 ways.

Original entry on oeis.org

107, 109, 116, 125, 140, 146, 168, 209, 249, 273, 297

Offset: 1

Robert Price, Nov 07 2017

Theorem: There are no further terms. Proof (from a proof by David A. Corneth on Nov 08 2017 in A294736): The von Eitzen link states that if n > 7845 then the number of ways to write n as a sum of 5 squares is at least 11. For n <= 7845 terms have been verified by inspection. Hence this sequence is finite and complete.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A025429, A025357, A294675, A294736.

fQ[n_] := Block[{pr = PowersRepresentations[n, 5, 2]}, Length@Select[pr, #[[1]] > 0 &] == 10]; Select[ Range@300, fQ](* Robert G. Wilson v, Nov 17 2017 *)

A295489 Numbers that have exactly six representations as a sum of six nonnegative squares.

Original entry on oeis.org

30, 33, 34, 35, 39, 40

Offset: 1

Robert Price, Nov 22 2017

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A294524, A295150.

A295153 Numbers that have exactly five representations as a sum of five nonnegative squares.

Original entry on oeis.org

29, 34, 35, 36, 38, 40, 41, 42, 44, 46, 55, 57

Offset: 1

Robert Price, Nov 15 2017

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000174, A006431, A294675.

A295485 Numbers that have exactly two representations as a sum of six nonnegative squares.

Original entry on oeis.org

4, 5, 6, 8, 10, 11, 15

Offset: 1

Robert Price, Nov 22 2017

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A294524, A295150.

A295486 Numbers that have exactly three representations as a sum of six nonnegative squares.

Original entry on oeis.org

9, 12, 13, 14, 16, 19, 23

Offset: 1

Robert Price, Nov 22 2017

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A294524, A295150.

A295487 Numbers that have exactly four representations as a sum of six nonnegative squares.

Original entry on oeis.org

17, 18, 22, 24, 31

Offset: 1

Robert Price, Nov 22 2017

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A294524, A295150.

A295488 Numbers that have exactly five representations as a sum of six nonnegative squares.

Original entry on oeis.org

20, 21, 25, 26, 27, 28, 32

Offset: 1

Robert Price, Nov 22 2017

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A294524, A295150.

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A294741 Numbers that are the sum of 5 nonzero squares in exactly 7 ways.

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A294742 Numbers that are the sum of 5 nonzero squares in exactly 8 ways.

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A294743 Numbers that are the sum of 5 nonzero squares in exactly 9 ways.

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A294744 Numbers that are the sum of 5 nonzero squares in exactly 10 ways.

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A295489 Numbers that have exactly six representations as a sum of six nonnegative squares.

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A295153 Numbers that have exactly five representations as a sum of five nonnegative squares.

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A295485 Numbers that have exactly two representations as a sum of six nonnegative squares.

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A295486 Numbers that have exactly three representations as a sum of six nonnegative squares.

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A295487 Numbers that have exactly four representations as a sum of six nonnegative squares.

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